What is Indefinite integral: Definition + 207 Threads
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function f. This can be stated symbolically as F' = f. The process of solving for antiderivatives is called antidifferentiation (or indefinite integration), and its opposite operation is called differentiation, which is the process of finding a derivative. Antiderivatives are often denoted by capital Roman letters such as F and G.Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.
In physics, antiderivatives arise in the context of rectilinear motion (e.g., in explaining the relationship between position, velocity and acceleration). The discrete equivalent of the notion of antiderivative is antidifference.
I proceeded as follows
$$\int\frac{2(\sqrt3-1)(cosx-sinx)}{2(\sqrt3+2sin2x)}dx$$
$$\int\frac{(cos(\pi/6)-sin(\pi/6))(cosx-sinx)}{(sin(\pi/3)+sin2x)}dx$$
$$\frac{1}{2}\int\frac{cos(\pi/6-x)-sin(\pi/6+x)}{sin(\pi/6+x)cos(\pi/6-x)}dx$$
$$\frac{1}{2}\int cosec(\pi/6+x)-sec(\pi/6-x)dx$$
Which leads...
\frac{d^{2}}{dx^{2}}\int_{0}^{x}(\int_{1}^{sint}\sqrt{1+u^{4}}du)dt=\frac{d}{dx}\int_{0}^{sinx}(\sqrt{1+u^{4}})du
then we let m=sinx，so x=arcsinx，then we get \frac{d}{dx}\int_{0}^{sinx}(\sqrt{1+u^{4}})du=\frac{dm}{dx}\frac{d}{dm}\int_{0}^{m}(\sqrt{1+u^{4}})du=\sqrt{1+m^{4}}\frac{dm}{dx}，then we...
When I encountereD this kind of question before.For example
\int x\sqrt{2+x^{2}}dx
We make the Substitution t=x^{2}+2，because its differential is dt=2xdx，so we get \int x\sqrt{2+x^{2}}=1/2\int\sqrt{t}dt，then we can get the answer easily
But the question，it seems that I can't use the way to...
Now the steps to solution are clear to me...My interest is on the constant that was factored out i.e ##\frac{2}{\sqrt 3}##...
the steps that were followed are; They multiplied each term by ##\dfrac{2}{\sqrt 3}## to realize,
##\dfrac{2}{\sqrt 3}\int \dfrac{dx}{\left[\dfrac{2}{\sqrt...
I have the following integration -
$$\int \frac{2}{x - b \frac{x^{m - n + 1}}{(-x + 1)^m}} \, dx $$
To solve this I did the following -
$$\int \frac{1 - b \frac{x^{m - n}}{(-x + 1)^m}+1 + b \frac{x^{m - n}}{(-x + 1)^m}}{x(1 - b \frac{x^{m - n}}{(-x + 1)^m})} \, dx $$
Which gives me -...
Problem statement : I start by putting the graph of (the integrand) ##f(x)## as was given in the problem. Given the function ##g(x) = \int f(x) dx##.
Attempt : I argue for or against each statement by putting it down first in blue and my answer in red.
##g(x)## is always positive : The exact...
Hi
When we find integrals of Bessel function we use recurrence relations.
But this requires that we have the variable X raised to some power and multiplied with the function .
But how about when we have Bessel function of first order and without multiplication?
How should we integrate it ?
Since ##h## and ##k## are constants:
$$\frac{h}{k}\cdot \int \frac{1}{y(h-y)} \ dy$$
Now, rewriting the integrating function in terms of coefficients ##A## and ##B##:
$$\frac{1}{y(h-y)}=\frac{A}{y}+\frac{B}{h-y}\rightarrow B=A=\frac{1}{h} \rightarrow$$
$$\frac{1}{h}\int \frac{1}{y}\ dy +...
Hi guys,
I got to solve this integral in a recent test, and literally I had no idea of where to start.
I thought about substituting ##tan(\frac{x}{2})=t## in order to apply trigonometry parametric equations, integrating by parts, substituting, but I always found out I was just running in a...
That's my attempt:
$$\int (\frac{1}{cos^2x\cdot tan^3x})dx = \int (\frac{1}{cos^2x}\cdot tan^{-3}x) dx$$
Now, being ##\frac{1}{cos^2x}## the derivative of ##tanx##, the integral gets:
$$-\frac{1}{2tan^2x}+c$$
But there is something wrong... what?
If we look at the denominator of this integral $$\int \frac{\cos x + \sqrt 3}{1 + 4\sin \left(x+ \pi/3\right) + 4\sin^2 \left(x+\pi/3\right)} dx$$ then we can see that ## 1 + 4\sin \left(x+ \pi/3\right) + 4\sin^2 \left(x+\pi/3\right) = \left(1+2\sin\left(x+\pi/3\right)\right)^2## and ##...
Homework Statement
The indefinite integral $$\int \, $$ and it's argument.
The indefinite integral has a function of e.g ## \cos (x^2) \ ## or ## \ e^{tan (x)} \ ##
If the argument of ## \cos (x^2) \ ## is ## \ x^2 \ ## then the argument of ## \ e^{tan(x)} \ ## is ## \ x \ ## or ## \ tan (x) \...
I have seen two approaches to the method of integration by substitution (in two different books). On searching the internet i came to know that Approach I is known as the method of integration by direct substitution whereas Approach II is known as the method of integration by indirect...
I was wandering if there is a way to understand whether it is possible to find an indefinite integral of a function. Let's say e^(-x^2) or e^(x^2)... They can't have indefinite integrals, but how can I say it? Is there a theorem or something?
Hi all! I'm new to Mathematica.
I have written a code for performing a convolution integral (as follows) but it seems to be giving out error messages:
My code is:
a[x_?NumericQ] := PDF[NormalDistribution[40, 2], x]
b[k_?NumericQ, x_?NumericQ] := 0.0026*Sin[1.27*k/x]^2
c[k_?NumericQ...
Homework Statement
Calculate the indefinite integral of the function ## \int\frac{3x^3}{\sqrt{1-x^2}}##
my book gives the answer ##-(2+x^2)\sqrt{1-x^2}+C##
Homework EquationsThe Attempt at a Solution
So I started trying to calculate this indefinite integral by using a substitution...
Homework Statement
find the following integral:
cos(x/2) - sin(3x/2) dxHomework Equations
I think the substitution method has to be used.
Solve integrals by parts.
The Attempt at a Solution
Let u = x/2
cosu
du/dx=1/2, I then inverted it so dx/du = 2/1 = 2
So dx=2du
Now I have cosu2du
Do I...
F(x) = \int_a^x f(t) dt
I have found various arguments online for both.
Personally I think it's an indefinite integral because:
1) Its upper limit is a variable and not a constant, meaning the value of the integral actually varies with x. This is no different to the family of primitives...
Hello,
Please can someone help me solve my problem. I have recently submitted my answer and had my work referred for an error. I have pictures of my question and working out, however i can not seem to post them on the page. can i email them to someone for advice/guidance
Thanks
I have a calculator which allows me to sketch indefinite integrals - it assumes c = 0. However, when I try to use Desmos Online Graphing Calculator, it won't let me do this with it's integral function. It keeps trying to make me use definite integrals.
I know that ∫(a,b,f(x)dx = F(a) - F(b), so...
What is the $\displaystyle \begin{align*} \int{ \frac{54\,t - 12}{\left( t- 9 \right) \left( t^2 - 2 \right) } \,\mathrm{d}t } \end{align*}$
We should use Partial Fractions to simplify the integrand. The denominator can be factorised further as $\displaystyle \begin{align*} \int{ \frac{54\,t -...
Homework Statement
∫(sin2(πx)*cos5(πx))dx.
Homework Equations
Just the above.
The Attempt at a Solution
I have no idea how pi effects the answer, so I basically solved ∫(sin2(x)^cos5(x))dx.
∫(sin2(x)*cos4(x)*cos(x))dx
∫sin2(x)*(1-sin2(x))2*cos(x))dx
U-substitution
u = sin x du =...
Homework Statement
Use the Fourier transform to compute
\int_{-\infty}^\infty \frac{(x^2+2)^2}{(x^4+4)^2}dx
Homework Equations
The Plancherel Theorem
##||f||^2=\frac{1}{2\pi}||\hat f ||^2##
for all ##f \in L^2##.
We also have a table with the Fourier transform of some function, the ones of...
Homework Statement
okay so the equation goes:
∫(x*sin2(x))/(x3-1) over the terminals:
b= ∞ and a = 2
Homework Equations
Various rules applying to the convergence or divergence of integrals such as the p-test, ratio test, squeeze test etc
The Attempt at a Solution
Okay so I have tried...
First of all, I'm new here, so please bear with me if the answer to my question can be found elsewhere, but I have been working a problem and searching for an answer for two weeks now without a complete solution. In Eisberg and Resnick chapter 5, problem 15, an essential part of the problem is...
Homework Statement
Evaluate the Integral:
∫sin2x dx/(1+cos2x)
Homework EquationsThe Attempt at a Solution
I first broke the numerator up:
∫2sinxcosx dx /(1+cos2x)
2∫sinxcosx dx /(1+cos2x)
Then I let u = cosx so that du = -sinx dx
-2∫u du/(1+u2)
And now I'm stuck. I thought about turning...
Suppose ##f## is defined as follows:
##f(x) = 1## for all ##x ≠ 1##, ##f(1) = 10##.
Is the indefinite integral (or the most general antiderivative) of ##f## defined at ##x = 1##?
I'm asking this question because I already know how to deal with, say, ##\int_0^2 f##; ##f## has only one removable...
Homework Statement
Evaluate ∫e-θcos2θ dθ
Homework Equations
Integration by parts formula
∫udv = uv -∫vdu
The Attempt at a Solution
So in calc II we just started integration by parts and I'm doing one of the assignment problems. I know I need to do the integration by parts twice, but I've hit...
Just for fun, I tried this rather trivial problem, but I think I went wrong somewhere:
$$\int arcsec(x) \ dx$$
Let ##arcsec(x)=y## . Then ##x=sec \ y##, or ##y=arccos(\frac 1{x})##
So the problem becomes $$\int arccos(\frac 1 {x}) \ dx$$
Let ##\frac 1 {x} = cos \ u## , so that ##dx = secu \ tanu...
1.
http://www.imageurlhost.com/images/cnj1t05jh6e4fxqy4i5_integral.png
I know that this integral is solved by the sustitution method
The Attempt at a Solution
I tried converting the integral to the form of Arctanx, but that x2 on the numerator ruined everything. Thanks
I've been contributing to an open source calculator, and I wanted a way to take integrals of functions. I suppose you could implement a definite integral function by using Riemann Sums, but I can't find any way to implement indefinite integrals (or derivatives for that matter).
I've heard that...
Homework Statement
I was trying to find a general expression for the arc length of sine, i.e. ##\int \sqrt{1+cos^2 x} dx##, but got stuck. I made this problem up, and with some google searches realized that it uses something known as an elliptical integral. How do I go about using it here?
2...
Evaluation of Indefinite Integral $\displaystyle \int_{0}^{1} \sqrt{1-2\sqrt{x-x^2}}dx$
$\bf{My\; Try::}$ We can write the given Integral as
$\displaystyle \int_{0}^{1}\sqrt{\left(\sqrt{x}\right)^2+\left(\sqrt{1-x}\right)^2-2\sqrt{x}\cdot \sqrt{1-x}}dx$
So Integral Convert into...
Evaluation of $$\displaystyle \int\frac{5x^3+3x-1}{(x^3+3x+1)^3}dx$$$\bf{My\;Try::}$ Let $\displaystyle f(x) = \frac{ax+b}{(x^3+3x+1)^2}.$ Now Diff. both side w. r to $x\;,$ We Get$\displaystyle \Rightarrow f'(x) = \left\{\frac{(x^3+3x+1)^2\cdot a-2\cdot (x^3+3x+1)\cdot (3x^2+3)\cdot...
Hello, everyone
I've trying to solve this integral but it seems like the methods I know are not enogh to solve it. So I'd be glad if you could give me some trick to get into the answer.
Here it is:
Thanks in advance!
The big clue here is the square root in the denominator, because $\displaystyle \begin{align*} \frac{\mathrm{d}}{\mathrm{d}x} \left( \sqrt{x} \right) = \frac{1}{2\,\sqrt{x}} \end{align*}$. So this suggests that you probably need to find a square root function to substitute. Rewrite your...
Is possible to compute indefinite integrals of functions wrt its variables, but is possible to compute indefinite integrals of scalar fields and vector fields wrt line, area, surface and volume?